Preflop: The problem with such min-raises is that that so many people with those small stacks are slowplaying hands like KK+ with their min-raise and it's somewhat dangerous. Since if we Call here we as well have to plan go broke postflop with TP against them and their stack. Therefore sometimes I might even just avoid such spots and fold with KQ and rather prefer to have at least KQs. Of course we are getting very good odds but then again we don't know him, maybe he really is doing that min-raise just with trap and we will never know where we stand with our Q or K when we do hit.

As playedPostflop: Not much to say. I might even just Call on turn and if he checks on river then ship it. There aren't that many draws anyways. Although shipping on turn is fine as well against some station.